ARTICLE

Received 19 Dec 2014 | Accepted 26 Oct 2015 | Published 3 Dec 2015

Ultralight shape-recovering plate mechanicalmetamaterialsKeivan Davami1, Lin Zhao2, Eric Lu3, John Cortes1, Chen Lin1, Drew E. Lilley3, Prashant K. Purohit1 & Igor Bargatin1

Unusual mechanical properties of mechanical metamaterials are determined by their carefully

designed and tightly controlled geometry at the macro- or nanoscale. We introduce a class of

nanoscale mechanical metamaterials created by forming continuous corrugated plates out of

ultrathin films. Using a periodic three-dimensional architecture characteristic of mechanical

metamaterials, we fabricate free-standing plates up to 2 cm in size out of aluminium oxide

films as thin as 25 nm. The plates are formed by atomic layer deposition of ultrathin alumina

films on a lithographically patterned silicon wafer, followed by complete removal of the silicon

substrate. Unlike unpatterned ultrathin films, which tend to warp or even roll up because of

residual stress gradients, our plate metamaterials can be engineered to be extremely flat.

They weigh as little as 0.1 g cm� 2 and have the ability to ‘pop-back’ to their original shape

without damage even after undergoing multiple sharp bends of more than 90�.

DOI: 10.1038/ncomms10019 OPEN

1 Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA. 2 NanotechologyMaster’s Program, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA. 3 Vagelos Integrated Program in Energy Research, University ofPennsylvania, Philadelphia, Pennsylvania 19104, USA. Correspondence and requests for materials should be addressed to I.B. (email:bargatin@seas.upenn.edu).

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Advances in micro/nanofabrication techniques recentlyresulted in demonstrations of macroscopic solids madesolely out of free-standing films with nanoscale

thickness1. These mechanical metamaterials have a carefullyengineered structure at the micro- and nanoscales, with periodicgeometries that lead to unusual mechanical properties. Forexample, mechanical metamaterials can be engineered to have arecord high stiffness for a given effective density or have a verylarge range of elastic deformation1–5. The examples reported inliterature typically have a truss-like structure with the underlyingmaterial forming an interconnected periodic framework that iseasily penetrated by air or other ambient. Such ‘bulk’metamaterials can be fabricated to occupy a macroscopicvolume in all three dimensions2, subject only to the limitationsof the used fabrication method. However, they cannot bythemselves be formed into a continuous plate that could serve,for example, as the wing of a flying microrobot.

Mechanical metamaterials are only the most recent develop-ment in the ongoing quest for materials with high stiffness andlow density. In fact, they can be viewed as a sub-class of cellularmaterials6, which have long been used in many applications,ranging from thermal insulation7 to aerospace structuralelements8. Aerogel materials are some of the oldest and mostcelebrated cellular materials due to their ultralow weight andthermal conductivity. Since their discovery in the 1930s (ref. 9),much effort has been dedicated to tailoring aerogels’ propertiesfor various applications. Aerogels with ultralow densities(1 mg cm� 3 or less) have been fabricated from silica, aluminaand carbon10–13; however, such aerogels are typically very brittle,with tensile, compressive and shear strengths in the order of1 MPa or less14, which significantly limits their applications15.More recently, a number of low-density carbon-based materialscomposed of sheets as thin as a single atomic layer16–19 have beendemonstrated. These often have unique mechanical properties;however, as with aerogels, the micro/nanoscale geometry of thesecellular materials can only be partly controlled and is largelyrandom.

Typically, the effective Young’s modulus, Eeff, of a cellularstructure scales as Eeff ¼ Es r�=rsð Þb, where r�=rs is the relativedensity, equal to the ratio of the apparent density of the cellularmaterial to that of the underlying solid film (that is, cell wall)material, Es, is the Young’s modulus of the cell wall material, andthe power-law exponent b is determined by the dominantdeformation mode in the cellular material. In materials withbending-dominated deformation, such as aerogels and foams,there is a quadratic or stronger relationship between the Young’smodulus and density, implying that material properties degradevery quickly as the apparent density is reduced6. However, it ispossible to obtain a lower exponent b by carefully engineering thearchitecture of the constituent elements of the cellular material20.Some of the recently reported truss-like metamaterials exhibit theideal linear relationship (b¼ 1) between the stiffness and density4,allowing them to occupy new areas4,5 on the stiffness-densityAshby chart21. Other truss-like mechanical metamaterialsfabricated out of ultrathin films (o100 nm) featured exponentsb between 1 and 2, but were surprisingly robust, recovering theiroriginal shape after deformations of up to 50% even whenmanufactured out of brittle materials such as alumina5. Incontrast to random cellular architecture typical of foams,nanoscale mechanical metamaterials can therefore have superiorstiffness and strength at very low densities because the constituentmaterials are distributed more regularly and efficientlythroughout the structure.

In addition to bulk mechanical metamaterials, periodic cellulararchitectures have been used extensively in the design of stiff andlightweight plates. This is typically achieved by patterning the

plate in the normal direction, that is, perpendicular to the planeof the sheet. For example, simple corrugated sheets consisting of asingle three-dimensional patterned layer can have significantlyenhanced bending stiffness22 and have found a wide range ofapplications from macroscale architecture23 to nanoscalesensors24 and energy devices25. Honeycomb lattices andsandwich structures, which consist of two face sheets attached toa periodic cellular core, have become ubiquitous in construction,aerospace, scientific instrumentation (for example, optical tables)and other industries that require lightweight rigid plates26,27.Although sandwich structures possess a high bending stiffness,they typically cannot sustain a sharp bend without fracture.

Here, we use microscale periodic cellular design to createrobust, stiff, and ultra-lightweight plates out of ultra-thin films created by atomic layer deposition (ALD). Similar totruss-like ‘bulk’ metamaterials, our plate metamaterials cansustain extreme deformations when made out of ALD films withthicknesses below 100 nm; although truss-like metamaterials canrecover after large compression or shear deformations, platemetamaterials can recover from extreme bending deformations.In contrast to truss-like bulk metamaterials, plate metamaterialsachieve such robustness while remaining geometrically conti-nuous, that is, without any structural holes. In addition, platemetamaterials can be fabricated in high volumes and withrelatively large lateral dimensions using standard opticallithography techniques. While even thinner suspended filmshave been made out of graphene and other two-dimen-sional materials28–30, the resulting membranes are oftenprone to tearing, warping or crumpling. As a result, suchmembranes are typically much less than 1 mm in lateraldimensions29,30 and remain flat only if put under tension on arigid frame. In contrast, our free-standing plates are completelystandalone and mechanically robust even at truly macroscalelateral dimensions (41 cm).

ResultsFabrication and geometry of honeycomb plate metamaterials.Figure 1a shows a cantilevered (single-clamped) rectangular platebased on a hexagonal honeycomb metamaterial design. The plateswere fabricated out of ALD alumina films with uniform thickness,which varied between 25 and 100 nm in different fabrication runs,using optical lithography and a combination of wet and dryetching techniques (see the Supplementary Note 1, Supplemen-tary Figs 1 and 2, and (ref. 31) for detail of the fabricationprocess). Alumina material was chosen as the cell wall materialfor its high stiffness, chemical and high-temperature resistance,as well as the highly conformal and pinhole-free nature of itsALD films.

As shown in Fig. 1b,c, the plate is able to maintain its shapebecause of its three-dimensional honeycomb microstructure,which increases the flexural stiffness of the plate by one to threeorders of magnitude depending on the used cell parameters (seeSupplementary Note 3 for detail). When bending stresses areapplied, rigidity is provided by the vertical walls and thehorizontal segments between the ‘hexagonal cups’ and the ‘cup’bottoms similar to the way that C-beams or corrugated sheetsresist deformation. This keeps the sheet flat even when the filmhas significant internal stresses due to fabrication-induced stressgradients that generally cause unpatterned films to curl up with asmall radius of curvature (Fig. 1d).

Robustness of plate metamaterials. In contrast to macroscalehoneycomb sandwich structures, our plates can sustain largebending deformations because these top and bottom surfaces arenot continuous, allowing the structures to fold with a very smallradius of curvature without fracture, as shown in Fig. 1e. Despite

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the brittleness of the cell wall ceramic material (aluminiumoxide), these ultrathin structures are robust and typically recovertheir original shapes even after undergoing extreme bendingdeformations multiple times (Supplementary Movie 1). Aswith previously reported bulk metamaterials5, structuresmade out of thinner films were typically more robust, that is,able to withstand sharper bends. This is simply because of thefact that the maximum strain along the fold lines formed inthese plates at large bending deformations is proportional tothe thickness of the solid cell wall film. In this aspect, platemetamaterials are very different from plates with effectivelyuniform cross-sections, such as corrugated sheets or sandwichplates, for which the classical plate theory holds and themaximum strains are therefore proportional to the curvaturemultiplied by the height of the plate rather than the thickness ofthe face sheets. If a sandwich plate undergoes a sharp bend(that is, with radius of curvature similar to the plate height), theinduced strains are of the order of one, which would lead tofailure of most structural materials. This remains true forsandwich plates made of films with macro-, micro- ornanoscale thickness. In contrast, the maximum strains in ourplates are determined by the ratio of film thickness to plateheight, which is in the order of 10� 3 for the thinnest ALD plateswe fabricated, explaining their observed robustness.

The design of the particular features and dimensions of theplates was guided by the results of COMSOL finite elementsimulations (see Supplementary Note 3). Based on the results ofthe simulations, we have focused our experiments on plates withhexagonal honeycomb geometry because of their approximatelyisotropic bending stiffness for small deformations6,32. Otherperiodic geometries are possible, but to achieve high flatness andbending stiffness, it is crucial that any plane perpendicular to theplate must intersect vertical walls (Fig. 1f). Otherwise, the planewill bend very easily along a line that contains only horizontalelements. Among regular polygon tiling patterns, only hexagonal(honeycomb) patterns can satisfy this requirement. Other, lower-symmetry tiling patterns, such as basket-weave or rhombille, alsosatisfy this requirement, but generally have a lower bendingstiffness (see Supplementary Note 3, Supplementary Fig. 7, andSupplementary Table 1).

Enhanced bending stiffness of plate metamaterials. Traditionalprismatic corrugated sheets with a periodic cross-section (Fig. 2a)offer a greatly enhanced bending stiffness in one direction butalmost no change in the bending stiffness in the perpendiculardirection22,23. The enhancement factor, EF, which can be used todescribe the increase in bending stiffness relative to a completely

0

200

400

600

0 10 20

Enh

ance

men

t, E

Fm

ax

Cell diameter/rib width, D/w

ba c

Dw

Figure 2 | Simulated bending properties of traditional corrugated sheets and plate metamaterials. (a) Illustration of the anisotropic bending stiffness of

periodic corrugated sheets with a constant cross-section. (b) Isotropic enhancement factor of honeycomb corrugated sheets versus the ratio of cell

diameter to rib width. (c) COMSOL finite element simulation illustrating the concentration of elastic energy in the areas where the ribs intersect. The colour

represents the elastic energy density, with red areas corresponding to the highest elastic energy density and blue to the lowest.

c

ed f

ba

Rib width

Cell diameter

Cell height

Thickness

Figure 1 | Geometry of honeycomb plate metamaterials. (a) An scanning electron microscopic (SEM) image of a 1-mm-long cantilevered plate made

using a hexagonal honeycomb metamaterial geometry. The 35-nm-thick ALD alumina film is coloured for clarity. Scale bar, 100mm. (b) An SEM image of

the detail of the cell structure. Scale bar, 10mm. (c) A schematic representation showing an individual cell with its characteristic dimensions. (d) An SEM

image of a 50-nm-thick cantilever without the honeycomb pattern, illustrating the warping typical of suspended flat ultrathin films. Scale bar, 200mm.

(e) An SEM image of a 1-mm-long 25-nm-thick plate bent by more than 90� using a micromanipulator without any fracture. After the tip of the

micromanipulator was removed, the plate recovered its original shape (see Supplementary Movie 1, Supplementary Note 2 and Supplementary Figs 3

and 4). Scale bar, 200mm. (f) Schematic illustrating the requirement that any plane perpendicular to the plate will intersect vertical walls. The insets show

zoomed-in detail of how the plane intersects individual cells.

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planar sheet, is determined by the moment of inertia of itsconstant cross section. Typically, it scales quadratically with theratio of height of the corrugation, h, to the thickness of the sheet,t, and depends only weakly on the other geometric parameters(for example, period) of the corrugation pattern: EFcorr � h

t

� �2.

In contrast, the hexagonal corrugation used in this work offersan enhanced bending stiffness in all directions, and its isotropicenhancement factor depends on all parameters of the unit cell.We have studied this dependence through extensive finiteelement simulations of small-deformation response in COMSOL(see Supplementary Note 3 and Supplementary Figs 5 and 6).Briefly, the enhancement factor of ultrathin plates initiallyincreases quickly with height, just like in simple corrugatedsheets, but then saturates for heights much greater than the filmthickness. This maximum saturated value of the enhancementfactor depends on the ratio of two in-plane parameters of the cellgeometry: EFmax � D

w

� �2, where D is the inner diameter of the

hexagons and w is the width of the ribs (Fig. 2b).An elastic energy analysis can be used to explain these results

for the maximum enhancement factor, as well as the overalldependence of the bending stiffness on the height of ourhoneycomb plates (see Supplementary Note 3 for detail). Briefly,for sufficiently large heights, the hexagons with their attachedvertical ribs become very rigid, and practically do not bend. As aresult, when one of our honeycomb plates is bent, the majority ofthe elastic energy is concentrated in the areas near the hexagonvertices, where the strengthening ribs intersect. Figure 2c showsone representative case, where B95% of the total elastic energy isconcentrated in these triangular regions according to COMSOLsimulations. Thus, the total area undergoing bending in ahoneycomb plate is only a small fraction of the total area, andthe bending stiffness is enhanced proportionally to the ratio ofthese areas.

More specifically, consider a unit cell of area A of a honey-comb patterned plate. Let the area occupied by the smalltriangular regions (near each vertex) per unit cell be A2 andtherefore A2¼A�A1 is the area of the remaining part of the unitcell. The corresponding mean curvature bending moduli ofthese regions are K1 and K2, respectively. If a uniformisotropic moment per unit length m is acting on the unit cell,then the elastic energy stored in the patterned plate per unit cellcan be written as U ¼ m2

2K1A1þ m2

2K2A2. Equating U ¼ m2

2KeffA,

where Keff is the effective (average) bending modulus for theplate metamaterial, we obtain AK � 1

eff ¼ A1K � 11 þA2K � 1

2 .

Now, A ¼ffiffi3p

2 Dþwð Þ2 and A2 ¼ffiffi3p

2 w2. Inthe limit of large plate heights, K1-N, and the effectivebending modulus is given by Keff ¼ A

A2K2, which gives

EFmax ¼ KeffK2� D

w þ 1� �2� D

w

� �2for D � w. The latter agrees

with the enhancement factor obtained by numerical simulations.More detailed analysis can be found in Supplementary Note 4 andSupplementary Fig. 8.

Measurements of bending stiffness. To validate the finite ele-ment model used for metamaterial designs and determine theYoung’s modulus of the cell wall ALD alumina material, we havemeasured the small-deformation bending stiffness of honeycombplates using an atomic force microscope (AFM). In particular, thespring constants of the cantilevered plates were inferred from theforce-displacement measurements obtained using an AFM.

Somewhat unexpectedly, the measured stiffness of thehoneycomb cantilevers depends very strongly on whethercontinuous vertical walls are present on the sides of thecantilevers (compare Fig. 3a,b). Such walls can be formedat cantilever edges either intentionally by designing them into

the lithography mask or unintentionally due to fabricationimperfections. When sidewalls are absent, the cantilever’s springconstant k ¼ 3WcantKeff=L3 is directly proportional to thebending stiffness of the metamaterial, Keff, introduced in theprevious section, and the cantilever width Wcant. As a result, thespring constant scales cubically with the film thickness,kpt3 D

w

� �2; and is almost independent of the height of the

structure (Fig. 3c), as discussed in the previous section. Incontrast, when sidewalls are present, the force-displacementresponse of the cantilevers is dominated by the stiffness of the twoC-beams formed by these walls at each side of the cantilever. As aresult, the corresponding spring constants scales linearly withthickness and quadratically with plate height, kph2t, as expectedfrom the stiffness scaling of conventional C-beams andcorrugated sheets (Fig. 3d). Figure 3e compares these experi-mental results to the spring constants from finite elementsimulations for plates with a cell height of 10 mm, a cell diameterof 50 mm, and ALD alumina thickness between B25 andB100 nm. The finite element simulations agree with experi-mental results best if we assume a Young’s modulus of 130 GPa,which is within the range of elastic moduli reported for ALDalumina films in the literature33.

To illustrate the repeatability and stability of the elasticbehaviour of the ALD honeycomb plates, we also performedcyclic loading tests: a plate with a honeycomb depth of 10mm andan ALD layer thickness of 72 nm was bent and unloaded 400times consecutively without exhibiting any significant changes inits spring constant (see Supplementary Note 5 and Supplemen-tary Fig. 9 for detail).

Properties of centimetre-scale plates. The robustness and flat-ness of the honeycomb periodic metamaterial allowed us tofabricate honeycomb plates on the macroscopic scale with lateraldimensions on the centimetre scale (Fig. 4). While the ALD layerthickness is only B50 nm, the devices do not show any visiblewarping or deformation. The plates fully recover their shape afterbending (Fig. 4a,b). They are also extremely lightweight (Fig. 4c)and float in air, descending very slowly (see SupplementaryMovie 2).

For the largest honeycomb plates we made (Fig. 4), the cellparameters shown in Fig. 1c must be simultaneously optimizedfor high bending stiffness, robustness to defects, and ease offabrication. In particular, we fixed the minimum feature size at10 mm to maximize the yield of optical lithography and etchingfabrication steps, and that determined the rib width w¼ 10 mm inall fabricated samples. While the bending stiffness increases withincreasing cell diameter (see Supplementary Note 3), we limitedcell diameters to D¼ 50mm in fabricated samples to maximizethe robustness of the films. Plates using cells with larger diametersare more prone to damage when probed with a micromanipulatortip or when individual fabrication defects occur becausedamaging a single cell affects a significant portion of the totalstructure. In contrast, in our fabricated plates, the fabricationdefects or external damage due to, for example, a puncture by anAFM tip, was localized to within a few cells because the crackstypically terminated at the vertical walls (see Supplementary Note6 and Supplementary Figs 10–12 for detail).

The cell height, which can be viewed as the effective thicknessof the plate metamaterial, was varied between 1 and 14 mm,corresponding to a range of etching depths easily accessible withstandard semiconductor reactive ion etching (RIE) tools. Theprimary effect of increasing the cell height is an increase in thebending stiffness and flatness (see Supplementary Note 3).The plates with the largest heights of 10� 14 mm were remarkablyflat, with the typical vertical displacement of only ±4mmover the 0.8� 1.2 mm field of view of the profilometer (Fig. 5a).

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The measured radii of curvature were in the order of 10 cm(Fig. 5b,c).

DiscussionThe mass per unit area (areal density) of our plate metamaterialscan be easily determined from the geometry, film thickness and

density of the cell wall material. For the typical density of ALDalumina of B4 g cm� 3 (ref. 33), the mass per unit area isbetween 10 and 40 mg cm� 2 (0.1� 0.4 g m� 2). For comparison,common soap bubbles and films can be made with thicknesses onthe order of 100 nm and therefore areal densities of 0.1 g m� 2,but they are fluid and ephemeral, unable to maintain their shapedue to gravity-induced flow and evaporation34. The thinnest

ba c

Figure 4 | Photographs of large-area honeycomb plates. Before (a) and after (b) pictures of a bent 0.5� 1.0 cm honeycomb plate. (c) Same plate placed

on flower petals does not cause any bending of the petals. Scale bar, 3 mm. The inset zooms in on a section of the plate showing the honeycomb

microstructure.

0.25

a b

c d

e

40

30

20

10

00 100 200 300 400

0.20

0.15

For

ce, n

NS

prin

g co

nsta

nt, N

m–1

For

ce, n

N

0.10

0.05

0.00

10–2

10–3

10–4

10–5

20 40 60

Withoutsidewalls

With sidewalls

k ~ t 1.2 ± 0.3

k ~ t 2.9 ± 0.5

80 100Film thickness, nm

0 100 200 300 400

10 µm, 50 nm

10 µm, 75 nm

10 µm, 35 nm

10 µm, 25 nm

10 µm, 106 nm

10 µm, 72 nm

h = 7 µm, t = 100 nm

10 µm, 53 nm

h = 14 µm, t = 53 nm

Displacement, nm Displacement, nm

Figure 3 | Measured bending response of honeycomb cantilevers with and without sidewalls. Scanning electron micrographs of a 1-mm-long cantilevers

(a) without and (b) with vertical sidewalls, which dominate the spring constant of cantilever when present. Scale bar, 50mm in both micrographs. The

insets show the vertical sidewall detail. (c) Force-displacement curves measured for 1-mm-long ALD honeycomb cantilevered plates (without vertical

sidewalls) for different thicknesses and heights. Note that cantilevers with a thickness of B100 nm and heights of 7 and 10mm have essentially the same

spring constant. (d) Same for cantilevers with vertical sidewalls, which increase the overall stiffness of the cantilever by up to two orders of magnitude.

Note that the cantilevers with a thickness of B50 nm and heights of 10 and 14mm have dramatically different spring constants, in accordance with the

scaling kph2t.(e) A comparison between experimental results and simulations for honeycomb structures with 10 mm height and different ALD film

thicknesses (with and without sidewalls). The curves show the actual predictions of the finite-element simulations (solid lines) along with ±20%

confidence interval (dashed lines). The symbols show experimental data. The experimental errors bars denote s.e.m. in c,d and s.d. in e. For the top curve

(with sidewalls) in e, the vertical error bars are smaller than the symbols.

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large-area plastic sheets that are commonly available are 0.5-mm-thick sheets of Mylar (also known as ‘OS film’), which have anareal density on the order of 0.7 g m� 2, making them B10 timesthicker and three times heavier than the ALD plates shown inFig. 4. At the same time, the flexural stiffness of OS films isapproximately twice smaller than that of 50-nm-thick ALDhoneycomb plates (see Supplementary Note 8 for detail). As aresult, these Mylar films wrinkle easily, sag under their ownweight, and generally cannot maintain their shape unless put on aframe under tension (see Supplementary Figs 14 and 15).

The ALD plates we have fabricated have very high aspectratios: height-to-thickness ratio of up to B1,000 and the length-to-height ratio of up to B1,000 as well, for an overall length-to-thickness aspect ratio of up to one million. As mentioned earlier,these large height-to-thickness aspect ratios are responsible forthe bending robustness of our plates, but such aspect ratios wouldbe very difficult to realize using conventional materials at themacroscale. To illustrate the extreme aspect ratios used to achievethe unique mechanical properties of these structures, it is usefulto compare them to origami paper structures, from which somerecently reported mechanical metamaterials were made35,36. If amodel of our 50-nm-thick ALD plates (Fig. 4) was made out ofstandard 100-mm-thick paper (that is, scaled up by a factor of2,000), the height of the corrugated paper plate would need to be10 mm� 2,000¼ 20 mm, and the lateral dimensions would be10� 20 m—about the size of a standard tennis court. Such largefree-standing paper structures obviously cannot maintain theirshape under gravity. Only by using cell walls with nanoscalethickness made from a stiff material like alumina we were able tofabricate free-standing plates with such large aspect ratios.

In summary, we have demonstrated shape-recovering, flat,ultrathin, continuous plate metamaterials with enhanced bendingstiffness. Similar to the recently reported ‘bulk’ mechanicalmetamaterials4, these plates are extremely lightweight andresilient because of their nanoscale thickness and microscalecellular structure, but they also form flat continuous plates. Theunusual properties of the plate metamaterials stem from boththeir hexagonal corrugation geometry, which enhances flatnessand bending stiffness, and the extremely low thickness of the film,which reduces weight and increases robustness. They are amongthe thinnest and lightest free-standing macroscale solids evermade, that is, objects that are chemically stable, able to maintaintheir shape, and be handled with bare hands.

The continuous nature of the plates, their robustness, and verylow weight make them suitable for a number of applications. Forexample, these plates can be used to make thinner wings thananything created by nature or mankind so far for applications inflying microrobots or microstructures37,38. The high Young’s

modulus and the refractory properties of the cell wallalumina material enable such applications even in harshenvironments. The extremely low thickness and weight mayallow such mechanical metamaterials to serve as acoustic oroptical metamaterials as well. Finally, our plate metamaterials alsooffer a new approach for measuring the mechanical properties ofengineered nanoscale materials: Thanks to the large lateraldimensions of the fabricated nanostructured plates, the materialproperties of these nanoscale materials can be studied at themacroscale using standard test methods and systems, such asInstron materials testing machines.

MethodsFabrication process. The fabrication started with a double side polished Si wafer.SiN films with a thickness of 180 nm were deposited on both sides using plasma-enhanced chemical vapor deposition (PECVD). A hexagonal lattice of regularhexagons (honeycomb structure) with a depth of up to 14 mm were patterned onthe front side of the silicon wafer using photolithography and RIE. The back sidewas patterned via photolithography and RIE etching of SiN to create openings forsubsequent release. Next, the SiN mask was removed from the front side using RIEetching and the ALD layer was then deposited using trimethylaluminium andwater precursors. The deposition rate was measured using an ellipsometer to be1.18 Å per cycle at 250 �C. Supplementary Fig. 1a shows a die with all three types ofmillimetre-scale device fabricated out of plate metamaterial: cantilevers, doublyclamped beams, and rectangular plates clamped on all four sides, whileSupplementary Fig. 1b shows one centimetre-scale cantilever on wafer. The sche-matic of the fabrication procedure is outlined in Supplementary Fig. 1c.

To pattern the ALD layer, a thick layer of SPR 220 resist (MicroChem Corp.)was spin-coated on the structure. After the spin coating and soft baking at 105 �C,the wafer was cooled down slowly to make sure the photoresist did not crack. Afterphotolithography, inductively coupled plasma etching with a BCl3-based chemistryprocess was used to pattern the alumina ALD layer. Next, most of the silicon underthe ALD plate was removed using anisotropic KOH etching. Before placing thewafer in KOH, the top surface was covered with ProTEK B3 (Brewer Science Inc.)to prevent the ALD layer from being etched in the KOH solution. Also, sincePECVD SiN does not cover the edge of the wafer, a protective chuck was used tokeep the edge inaccessible. A silicon etching rate of 75 mm h� 1 was measured at80 �C in the 30% KOH solution. By accurately timing the KOH etching process, itwas possible to stop the process B20 mm from the top surface. The exact depth wasmeasured using a Zygo white-light optical profilometer. After that, the ProTEKlayer was removed and a 15-min oxygen plasma cleaning was performed to removeany remaining polymer residue. Isotropic dry XeF2 etching was used for the finalrelease of the structure. Approximately 100 cycles (30 s each) of XeF2 etching with aratio of 3.2:2 (XeF2:N2) was needed to completely release the structures.

Shape recovery after extreme deformation. To study the flexibility and shaperecovery property of the structure, we used a micromanipulator inside a focusedion beam microscope. Honeycomb cantilevers with different ALD layer thicknessesunderwent complex loading and deformation. The cantilever recovered its shapeafter each set of deformations without showing any damage. Structures withthinnest ALD layer films were able to withstand the highest level of deformationrepeatedly without damage. Supplementary Fig. 3 illustrates a set of sequentialimages of a cantilever with a thickness of 25 nm under different loading showing itsflexibility and shape recovery. The corresponding video is also available asSupplementary Movie 1.

–4

–2

0

–4

–2

0

0.4 1.2 0.4 1.00.2 0.6 1.00.8X, mm

Y, mm

0.41.20.2

0.40.6

0.8–5

5

0

cba

X, mm Y, mm0.80 0.6 0.8

Z, µ

mZ, µ

m

Z, µ

m

Figure 5 | Residual curvature of centimetre-scale plates. (a) Profilogram of a free-standing ALD honeycomb plate resting on a flat chuck, obtained using a

Zygo white-light optical profilometer. The vertical scale is greatly exaggerated (by approximately two orders of magnitude) to illustrate the small residual

curvature of the plate. For this large field of view of the profilometer, the sidewalls and bottom surfaces of the plate are not resolved correctly and therefore

do not appear in this plot (see Supplementary Fig. 13 and Supplementary Note 7 for higher-resolution profilograms that show sidewalls and cup bottoms).

(b) Cross-section profile and the parabolic fit to the top surfaces of the plate, used to determine the radius of curvature of the plate along the X direction

(rXE5 cm). The inset shows the location of the cross section. (c) Same for the perpendicular Y direction (rYE11 cm).

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Finite element simulations. The small-deformation response of plate meta-materials was simulated using the structural mechanics module of COMSOLMultiphysics (version 5.0). We assumed constant cell wall thickness and isotropiclinear elastic properties, with a Young’s modulus of 130 GPa and a Poisson ratio of0.22. Given the extreme aspect ratios of the plate metamaterial, we modelled thestructures using thin shell geometry with triangular mesh elements. Geometricnonlinearity was included but did not significantly affect the results for the rela-tively small deformations used in AFM testing (Fig. 2).

For the simulations of the bending stiffness of an infinite plate (SupplementaryFig. 5), pure bending moments were applied to the loaded ends of a unit cell, andmirror symmetry conditions to the remaining two sides. The bending stiffnessfrom the unit cell simulations was compared to the bending stiffness inferred fromthe simulated spring constant of cantilevered plates under a point load at the tip(corresponding to AFM measurements), which provided an estimate for theaccuracy of finite element simulations.

AFM measurements. AFM measurements of force-displacement curves wereperformed using an Asylum atomic force microscope at room temperatureand under ambient conditions. Two different types of AFM probes(NANOANDMORE, USA) were used for the characterization. For the honeycombcantilevers without the vertical sidewalls, the nominal spring constant of the AFMtip was B0.01 N m� 1, and the length, width and thickness of the cantilever were125mm, 34 mm and 350 nm, respectively. The measurements on the honeycombcantilevers with the vertical sidewalls were conducted using an AFM probe with anominal spring constant of 2 N m� 1 and a cantilever length of 225mm, width of27mm, and thickness of 2.7 mm. Before the measurement of the ALD plates, thespring constant of the tip, Ktip, was obtained using a thermal noise method.

After calculating the spring constant of the tip as well as obtaining the inverseoptical lever sensitivity (InvOLS), a central load was applied at the middle of thefree end of the cantilever using the tip. The load was in the range of 0.5–100 nNwhich is calculated by multiplying the trigger point (0.25 to 0.5 V), InvOLS(B100 nm V� 1), and the spring constant of the cantilever (0.01 or B2 N m� 1).The beam displacement can be calculated from dh¼DZp�DZc, wheredh, DZp and DZc are the honeycomb cantilever deflection, the AFM piezo traveldistance, and the AFM cantilever deflection, respectively. Knowing that1/Ktotal¼ 1/Ktipþ 1/Kbeam, and Ktotal is the slope of the linear section of the force-displacement graph, the spring constant of the beam can be calculated.

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AcknowledgementsWe thank John Bassani, Kevin Turner and Robert Carpick for their comments on themanuscript, Noah Clay and J. Provine for their advice on nanofabrication, Jamie Ford ofthe Nanoscale Characterization Facility and Matthew Brukman of Nano/Bio InterfaceCenter (NBIC) for their help with characterization, and Lars-Patrik Roeller for his helpwith some of the graphs. The use of the University of Pennsylvania’s Nano/Bio InterfaceCenter (NBIC) instrumentation is acknowledged. This work was supported, in part, bythe School of Engineering and Applied Science at the University of Pennsylvania and aseed grant of the Center of Excellence for Materials Research and Innovation (CEMRI),NSF grant DMR11-20901.

Author contributionsK.D., L.Z., E.L, J.C. and C.L. performed experimental work. K.D., L.Z., J.C. and I.B.analysed experimental results. E.L. and D.E.L. performed finite element simulations.P.K.P. performed analytical theoretical analysis. I.B. conceived, designed and directed theproject. K.D. and I.B. wrote the manuscript with an input from all co-authors.

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How to cite this article: Davami, K. et al. Ultralight shape-recovering plate mechanicalmetamaterials. Nat. Commun. 6:10019 doi: 10.1038/ncomms10019 (2015).

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